Question: The sum of two numbers is $76$, and their difference is $54$. What are the two numbers?
Answer: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 76}$ ${x-y = 54}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 130 $ $ x = \dfrac{130}{2} $ ${x = 65}$ Now that you know ${x = 65}$ , plug it back into $ {x+y = 76}$ to find $y$ ${(65)}{ + y = 76}$ ${y = 11}$ You can also plug ${x = 65}$ into $ {x-y = 54}$ and get the same answer for $y$ ${(65)}{ - y = 54}$ ${y = 11}$ Therefore, the larger number is $65$, and the smaller number is $11$.